306] THE GEOMETRICAL THEORY. 329 



of the screws of this system will have, as its instantaneous screw, a screw of 

 infinite pitch parallel thereto. We have thus a system of impulsive screws 

 of the third order, and a corresponding system of instantaneous screws of 

 the third order, the relation between each pair being quite independent of 

 whatever particular rigid body of the group the impulsive wrench be 

 applied to. 



This system of the third order taken in conjunction with the cylindroid 

 (?;, ) will enable us to determine the total system of impulsive screws which 

 possess the property in question. Take any screw 6, of zero pitch, passing 

 through the centre of gravity, and any screw, (f&amp;gt;, on the cylindroid (w, ). 

 We know, of course, as already explained, the instantaneous screws corre 

 sponding to 9 and &amp;lt;. Let us call them \, /j,, respectively. Draw the 

 cylindroid (6, &amp;lt;), and the cylindroid (X, //,). The latter of these will be the 

 locus of the instantaneous screws, corresponding to the screws on the former 

 as impulsive screws. From the remarkable property of the two cylindroids, 

 so related, it follows that every impulsive screw on (9, &amp;lt;) will have its 

 corresponding instantaneous screw on (X, p) definitely fixed. This will be so, 

 notwithstanding the arbitrary element remaining in the rigid body. From 

 the way in which the cylindroid (9, (f&amp;gt;) was constructed, it is plain that the 

 screws belonging to it are members of the system of the fifth order, formed 

 by combinations of screws on the cylindroid (rj, ) with screws of the special 

 system of the third order passing through the centre of gravity. But all 

 the screws of a five-system are reciprocal to a single screw. The five-system 

 we are at present considering consists of the screws which are reciprocal to 

 that single screw, of zero pitch, which passes through the centre of gravity 

 and intersects both the screws, of zero pitch, on the impulsive cylindroid 

 (77, ). The corresponding instantaneous screws will also form a system of the 

 fifth order, but it will be a system of a specialized type. It will be the result 

 of compounding all possible displacements by translation, with all possible 

 twists about screws on the cylindroid (a, /3). The resulting system of the 

 fifth order consists of all screws, of whatsoever pitch, which fulfil the single 

 condition of being perpendicular to the axis of the cylindroid (a, /3). Hence 

 we obtain the following theorem : 



If an impulsive cylindroid, and the corresponding instantaneous cylin 

 droid, be known, we can construct, from these two cylindroids, and without any 

 further information as to the rigid body, two systems of screws of the fifth 

 order, such that an impulsive wrench on a given screw of one system will 

 produce an instantaneous twist velocity about a determined screw on the other 

 system. 



It is interesting to note in what way our knowledge of but two corre 

 sponding pairs of impulsive screws and instantaneous screws just fails to 

 give complete information with respect to every other pair. If we take any 



